Modular Curves, Modular Surfaces, and Modular Fourfolds

Details Modular Curves, Modular Surfaces, and Modular Fourfolds Modular Curves, Modular Surfaces, and Modular Fourfolds

2006-09-16

arxiv.org/abs/math/0609459v1

Mathematics

Number Theory

15 pages, Proceedings of a conference on Algebraic Cycles and Motives (in honor of Jacob Murre) in Leiden, The Netherlands

Scientific paper

This article surveys some recent work of the author on Hilbert modular fourfolds X. After some preliminaries on the cohomology and special, codimension 2 cycles Z on X of Hirzebruch-Zagier type, a proof of the Tate conjecture for X over abelian fields is exposed. It is followed by a description of a method to construct classes in the Bloch's Chow group CH^3(X,1) by using Hecke translates of the cycles Z as above with suitable intersections in translates of modular curves. The article ends with the introduction of a modular Gersten complex for a general Shimura variety X and the corresponding groups CH^p_{mod}(X) and CH^p_{mod}(X,1).

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